3.54.73 \(\int \frac {405-648 x^2-288 x^3+216 x^4+48 x^6+128 x^7-48 x^8-48 e^{12} x^8+e^6 (72 x^4+96 x^6+128 x^7-96 x^8)}{16 x^6} \, dx\)

Optimal. Leaf size=35 \[ x-\frac {\left (-3+\frac {9}{4 x^2}-x-x \left (1-x-e^6 x\right )\right )^2}{x} \]

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Rubi [A]  time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.83, number of steps used = 4, number of rules used = 3, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6, 12, 14} \begin {gather*} -\frac {81}{16 x^5}-\left (1+e^6\right )^2 x^3+\frac {27}{2 x^3}+4 \left (1+e^6\right ) x^2+\frac {9}{x^2}+3 \left (1+2 e^6\right ) x-\frac {9 \left (3+e^6\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(405 - 648*x^2 - 288*x^3 + 216*x^4 + 48*x^6 + 128*x^7 - 48*x^8 - 48*E^12*x^8 + E^6*(72*x^4 + 96*x^6 + 128*
x^7 - 96*x^8))/(16*x^6),x]

[Out]

-81/(16*x^5) + 27/(2*x^3) + 9/x^2 - (9*(3 + E^6))/(2*x) + 3*(1 + 2*E^6)*x + 4*(1 + E^6)*x^2 - (1 + E^6)^2*x^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {405-648 x^2-288 x^3+216 x^4+48 x^6+128 x^7+\left (-48-48 e^{12}\right ) x^8+e^6 \left (72 x^4+96 x^6+128 x^7-96 x^8\right )}{16 x^6} \, dx\\ &=\frac {1}{16} \int \frac {405-648 x^2-288 x^3+216 x^4+48 x^6+128 x^7+\left (-48-48 e^{12}\right ) x^8+e^6 \left (72 x^4+96 x^6+128 x^7-96 x^8\right )}{x^6} \, dx\\ &=\frac {1}{16} \int \left (48 \left (1+2 e^6\right )+\frac {405}{x^6}-\frac {648}{x^4}-\frac {288}{x^3}+\frac {72 \left (3+e^6\right )}{x^2}+128 \left (1+e^6\right ) x-48 \left (1+e^6\right )^2 x^2\right ) \, dx\\ &=-\frac {81}{16 x^5}+\frac {27}{2 x^3}+\frac {9}{x^2}-\frac {9 \left (3+e^6\right )}{2 x}+3 \left (1+2 e^6\right ) x+4 \left (1+e^6\right ) x^2-\left (1+e^6\right )^2 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 63, normalized size = 1.80 \begin {gather*} -\frac {81}{16 x^5}+\frac {27}{2 x^3}+\frac {9}{x^2}-\frac {9 \left (3+e^6\right )}{2 x}+3 x+6 e^6 x+4 \left (1+e^6\right ) x^2-\left (1+e^6\right )^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(405 - 648*x^2 - 288*x^3 + 216*x^4 + 48*x^6 + 128*x^7 - 48*x^8 - 48*E^12*x^8 + E^6*(72*x^4 + 96*x^6
+ 128*x^7 - 96*x^8))/(16*x^6),x]

[Out]

-81/(16*x^5) + 27/(2*x^3) + 9/x^2 - (9*(3 + E^6))/(2*x) + 3*x + 6*E^6*x + 4*(1 + E^6)*x^2 - (1 + E^6)^2*x^3

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fricas [B]  time = 1.10, size = 69, normalized size = 1.97 \begin {gather*} -\frac {16 \, x^{8} e^{12} + 16 \, x^{8} - 64 \, x^{7} - 48 \, x^{6} + 216 \, x^{4} - 144 \, x^{3} - 216 \, x^{2} + 8 \, {\left (4 \, x^{8} - 8 \, x^{7} - 12 \, x^{6} + 9 \, x^{4}\right )} e^{6} + 81}{16 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(-48*x^8*exp(3)^4+(-96*x^8+128*x^7+96*x^6+72*x^4)*exp(3)^2-48*x^8+128*x^7+48*x^6+216*x^4-288*x^
3-648*x^2+405)/x^6,x, algorithm="fricas")

[Out]

-1/16*(16*x^8*e^12 + 16*x^8 - 64*x^7 - 48*x^6 + 216*x^4 - 144*x^3 - 216*x^2 + 8*(4*x^8 - 8*x^7 - 12*x^6 + 9*x^
4)*e^6 + 81)/x^5

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giac [B]  time = 0.18, size = 69, normalized size = 1.97 \begin {gather*} -x^{3} e^{12} - 2 \, x^{3} e^{6} - x^{3} + 4 \, x^{2} e^{6} + 4 \, x^{2} + 6 \, x e^{6} + 3 \, x - \frac {9 \, {\left (8 \, x^{4} e^{6} + 24 \, x^{4} - 16 \, x^{3} - 24 \, x^{2} + 9\right )}}{16 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(-48*x^8*exp(3)^4+(-96*x^8+128*x^7+96*x^6+72*x^4)*exp(3)^2-48*x^8+128*x^7+48*x^6+216*x^4-288*x^
3-648*x^2+405)/x^6,x, algorithm="giac")

[Out]

-x^3*e^12 - 2*x^3*e^6 - x^3 + 4*x^2*e^6 + 4*x^2 + 6*x*e^6 + 3*x - 9/16*(8*x^4*e^6 + 24*x^4 - 16*x^3 - 24*x^2 +
9)/x^5

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maple [A]  time = 0.08, size = 67, normalized size = 1.91

method	result	size
default	\(-x^{3} {\mathrm e}^{12}-2 x^{3} {\mathrm e}^{6}+4 x^{2} {\mathrm e}^{6}-x^{3}+6 x \,{\mathrm e}^{6}+4 x^{2}+3 x -\frac {81}{16 x^{5}}-\frac {72 \,{\mathrm e}^{6}+216}{16 x}+\frac {9}{x^{2}}+\frac {27}{2 x^{3}}\)	\(67\)
risch	\(-x^{3} {\mathrm e}^{12}-2 x^{3} {\mathrm e}^{6}+4 x^{2} {\mathrm e}^{6}+6 x \,{\mathrm e}^{6}-x^{3}+4 x^{2}+3 x +\frac {\left (-72 \,{\mathrm e}^{6}-216\right ) x^{4}+144 x^{3}+216 x^{2}-81}{16 x^{5}}\)	\(68\)
norman	\(\frac {-\frac {81}{16}+\left (4 \,{\mathrm e}^{6}+4\right ) x^{7}+\left (6 \,{\mathrm e}^{6}+3\right ) x^{6}+\left (-\frac {9 \,{\mathrm e}^{6}}{2}-\frac {27}{2}\right ) x^{4}+\left (-{\mathrm e}^{12}-2 \,{\mathrm e}^{6}-1\right ) x^{8}+\frac {27 x^{2}}{2}+9 x^{3}}{x^{5}}\)	\(71\)
gosper	\(-\frac {16 x^{8} {\mathrm e}^{12}+32 \,{\mathrm e}^{6} x^{8}-64 x^{7} {\mathrm e}^{6}-96 x^{6} {\mathrm e}^{6}+16 x^{8}-64 x^{7}+72 x^{4} {\mathrm e}^{6}-48 x^{6}+216 x^{4}-144 x^{3}-216 x^{2}+81}{16 x^{5}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(-48*x^8*exp(3)^4+(-96*x^8+128*x^7+96*x^6+72*x^4)*exp(3)^2-48*x^8+128*x^7+48*x^6+216*x^4-288*x^3-648*
x^2+405)/x^6,x,method=_RETURNVERBOSE)

[Out]

-x^3*exp(12)-2*x^3*exp(6)+4*x^2*exp(6)-x^3+6*x*exp(6)+4*x^2+3*x-81/16/x^5-1/16*(72*exp(6)+216)/x+9/x^2+27/2/x^
3

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maxima [A]  time = 0.36, size = 58, normalized size = 1.66 \begin {gather*} -x^{3} {\left (e^{12} + 2 \, e^{6} + 1\right )} + 4 \, x^{2} {\left (e^{6} + 1\right )} + 3 \, x {\left (2 \, e^{6} + 1\right )} - \frac {9 \, {\left (8 \, x^{4} {\left (e^{6} + 3\right )} - 16 \, x^{3} - 24 \, x^{2} + 9\right )}}{16 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(-48*x^8*exp(3)^4+(-96*x^8+128*x^7+96*x^6+72*x^4)*exp(3)^2-48*x^8+128*x^7+48*x^6+216*x^4-288*x^
3-648*x^2+405)/x^6,x, algorithm="maxima")

[Out]

-x^3*(e^12 + 2*e^6 + 1) + 4*x^2*(e^6 + 1) + 3*x*(2*e^6 + 1) - 9/16*(8*x^4*(e^6 + 3) - 16*x^3 - 24*x^2 + 9)/x^5

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mupad [B]  time = 3.51, size = 57, normalized size = 1.63 \begin {gather*} x^2\,\left (4\,{\mathrm {e}}^6+4\right )-x^3\,{\left ({\mathrm {e}}^6+1\right )}^2-\frac {\left (72\,{\mathrm {e}}^6+216\right )\,x^4-144\,x^3-216\,x^2+81}{16\,x^5}+x\,\left (6\,{\mathrm {e}}^6+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^4)/2 - (81*x^2)/2 - 18*x^3 - 3*x^8*exp(12) + 3*x^6 + 8*x^7 - 3*x^8 + (exp(6)*(72*x^4 + 96*x^6 + 128
*x^7 - 96*x^8))/16 + 405/16)/x^6,x)

[Out]

x^2*(4*exp(6) + 4) - x^3*(exp(6) + 1)^2 - (x^4*(72*exp(6) + 216) - 216*x^2 - 144*x^3 + 81)/(16*x^5) + x*(6*exp
(6) + 3)

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sympy [B]  time = 0.45, size = 70, normalized size = 2.00 \begin {gather*} - \frac {x^{3} \left (16 + 32 e^{6} + 16 e^{12}\right )}{16} - \frac {x^{2} \left (- 64 e^{6} - 64\right )}{16} - \frac {x \left (- 96 e^{6} - 48\right )}{16} - \frac {x^{4} \left (216 + 72 e^{6}\right ) - 144 x^{3} - 216 x^{2} + 81}{16 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(-48*x**8*exp(3)**4+(-96*x**8+128*x**7+96*x**6+72*x**4)*exp(3)**2-48*x**8+128*x**7+48*x**6+216*
x**4-288*x**3-648*x**2+405)/x**6,x)

[Out]

-x**3*(16 + 32*exp(6) + 16*exp(12))/16 - x**2*(-64*exp(6) - 64)/16 - x*(-96*exp(6) - 48)/16 - (x**4*(216 + 72*
exp(6)) - 144*x**3 - 216*x**2 + 81)/(16*x**5)

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